M Theory Lesson 64
Whilst on the topic of AdS/CFT, Michael Rios has an interesting post on dimension altering weak coupling phase transitions for N=4 SUSY Yang-Mills.
A continuous change in dimension from six down to five is reminiscent of Thurston's beautiful fractal 2-spheres, which are filled with a 1-dimensional curve. These arise in the study of 3-manifolds such as those with 1-punctured torus fibres over the circle. The punctures draw out a boundary for the manifold by tracing a knot. Now according to Matti, the fractional modular domains would fit into the domain for the once punctured torus moduli (the $n=2$ case) on the upper half plane. Perhaps the $n=5$ domain (or rather the theta functions) could be used to model a 5-sphere, much as the j-invariant Belyi map links the $n=2$ domain to $\mathbb{CP}^1$.
Note: For the new readers to this blog, our use of the term M Theory must not be confused with its more popular usage in string theory related papers. The letter M stands here possibly for Motive, or perhaps Monad. Although these terms do appear in the popular literature, they rarely correspond to the physical usage we would like to make of them.
A continuous change in dimension from six down to five is reminiscent of Thurston's beautiful fractal 2-spheres, which are filled with a 1-dimensional curve. These arise in the study of 3-manifolds such as those with 1-punctured torus fibres over the circle. The punctures draw out a boundary for the manifold by tracing a knot. Now according to Matti, the fractional modular domains would fit into the domain for the once punctured torus moduli (the $n=2$ case) on the upper half plane. Perhaps the $n=5$ domain (or rather the theta functions) could be used to model a 5-sphere, much as the j-invariant Belyi map links the $n=2$ domain to $\mathbb{CP}^1$.
Note: For the new readers to this blog, our use of the term M Theory must not be confused with its more popular usage in string theory related papers. The letter M stands here possibly for Motive, or perhaps Monad. Although these terms do appear in the popular literature, they rarely correspond to the physical usage we would like to make of them.
posted by Kea | 2:12 PM
7 Comments:
Dear Kea,
thanks for clarification of terms. Of course, although I cannot take M-theory in the conventional sense seriously as a physical theory, I feel great sympathy for those who work seriously to develop it or any another theory and it is very stimulating do exchange ideas.
What creates tensions is the refusal of string theorists to publicly respond to the criticism and admit some basic difficulties besides hype. And of course, the censorship in archives which emerged around second super string revolution is rather frustrating.
Yes, censorship has been a real concern, which is why we are so fortunate to live through the web revolution.
Thanks for the clarity in your definition of M-theory! ;-)
Yes, censorship has been a real concern, which is why we are so fortunate to live through the web revolution. - Kea
It's self-devaluating. Before the web everyone in a democracy was free to stand on a soap box at a street corner and lecture those passing if they wanted to hear (so long as they didn't cause too much of an obstruction in so doing). That was the definition of freedom.
Now it is the internet, doing the same thing. How many sites are there on the internet? The internet has not really changed things for the better. If anyone uses it to try to bypass censorship, no one listens.
Even Feynman's revolutionary work was censored out until he gave up. Tony Smith quotes the following about Feynman's censorship problems:
"... My way of looking at things was completely new, and I could not deduce it from other known mathematical schemes, but I knew what I had done was right.
... For instance,
take the exclusion principle ... it turns out that you don't have to pay much attention to that in the intermediate states in the perturbation theory. I had discovered from empirical rules that if you don't pay attention to it, you get the right answers anyway .... Teller said: "... It is fundamentally wrong that you don't have to take the exclusion principle into account." ...
... Dirac asked "Is it unitary?" ... Dirac had proved ... that in quantum mechanics, since you progress only forward in time, you have to have a unitary operator. But there is no unitary way of dealing with a single electron. Dirac could not think of going forwards and backwards ... in time ...
... Bohr ... said: "... one could not talk about the trajectory of an electron in the atom, because it was something not observable." ... Bohr thought that I didn't know the uncertainty principle ...
... it didn't make me angry, it just made me realize that ... [ they ] ... didn't know what I was talking about, and it was hopeless to try to explain it further.
I gave up, I simply gave up ...".
The above quotation is from The Beat of a Different Drum: The Life and Sciece of Richard Feynman, by Jagdish Mehra (Oxford 1994) (pp. 245-248).
If you look to see how Feynman's ideas eventually gained attention, it resulted from a long struggle of Dyson versus Oppenheimer, explained by Dyson in a video here (Dyson says Oppenheimer behaved like an "old bigot" and wouldn't listen at first).
... it didn't make me angry, it just made me realize that ... [they] ... didn't know what I was talking about, and it was hopeless to try to explain it further.
Thanks for the great Feynman quote, Nigel. But we live in a different time to Feynman. Physics needs a more radical change right now, and nobody who realises what that means can possibly give up trying to explain it.
Ummm. I think you're right. Another example, allegedly, of ignored genius was Weyl. I'm reading Weyl's The Theory of Groups and Quantum Mechanics, 2nd ed., 1930. It's kind of funny because it's the book mentioned in Not Even Wrong as being the one wrote with chapters on symmetry groups alternating with chapters on quantum theory: the joke is that only half the chapters were read. I can see why.
All the odd numbered chapters are pure maths (chapter 1: unitary geometry, chapter 3: groups and their representations, chapter 5: the symmetry permutation group and the algebra of symmetric transformations) while the even numbered chapters are straightforward physics (chapter 2: quantum theory, chapter 4: application of the theory of groups to quantum mechanics). The style is totally and obviously different in each case. Surprisingly, the physics chapters are brilliant.
The maths chapters begin with pages full of boring definitions that look hard to remember, easy to confuse, etc., while the physics chapters are exciting from the first line, stick to solid facts and are brilliant even in comparison to modern introductions to quantum mechanics. It's like two totally separate books, with the chapters shuffled together.
Some of the symmetry group theory is soaking in, but I won't get all I need from Weyl's book. Problem is, I write down a list of things I expect to find out from reading a book before starting to read, and always end up learning hardly any of the things I intended, but a whole load of unexpected gems. The funny thing is, historically all the most mathematical developments in physics have been experimentally guided. I can't think of one instance where maths is any more than a tool in empirically confirmed physics. Weyl couldn't use group theory to sort out forces because there wasn't any data on the weak or strong forces to reveal what the symmetry groups were in 1930. Yet the whole idea of using maths to usefully model physical facts is anathema to the string theorists, obsessed with unobservable speculation with spin-2 gravitons and supersymmetry. Sometime during the 60s or 70s people became disillusioned with the restrictions of reality and observable spacetime and decided there must be more to the universe than the observed number of dimensions, etc.
"These arise in the study of 3-manifolds such as those with 1-punctured torus fibres over the circle." If you mean by "3-manifolds" something like 3-d manifolds, then you're slipping into Weyl-type mathspeak, unless you're just writing for mathematicians.
Post a Comment
<< Home